Chapter 2 Measurements

CHAPTER OUTLINE Scientific Notation Error in Measurements Significant Figures Rounding Off Numbers SI Units Conversion of Factors Conversion of Units Volume & Density

What is a Measurement? quantitative observation comparison to an agreed upon standard every measurement has a number and a unit

A Measurement the unit tells you what standard you are comparing your object to the number tells you what multiple of the standard the object measures the uncertainty in the measurement

Scientists have measured the average global temperature rise over the past century to be 0.6°C °C tells you that the temperature is being compared to the Celsius temperature scale 0.6 tells you that the average temperature rise is 0.6 times the standard unit the uncertainty in the measurement is such that we know the measurement is between 0.5 and 0.7°C

A x 10n SCIENTIFIC NOTATION Scientific Notation is a convenient way to express very large or very small quantities. Its general form is A x 10n n = exponent coefficient 1  A < 10

SCIENTIFIC NOTATION To convert from decimal to scientific notation: Move the decimal point in the original number so that it is located after the first nonzero digit. Follow the new number by a multiplication sign and 10 with an exponent (power). The exponent is equal to the number of places that the decimal point was shifted. 7 5 0 0 0 0 0 0 7.5 x 10 7

Scientific Notation: Writing Large and Small Numbers A positive exponent means 1 multiplied by 10 n times. A negative exponent (–n) means 1 divided by 10 n times.

SCIENTIFIC NOTATION For numbers smaller than 1, the decimal moves to the left and the power becomes negative. 0 0 0 6 4 2 3 6.42 x 10

1. Write 6419 in scientific notation. Examples: 1. Write 6419 in scientific notation. decimal after first nonzero digit power of 10 64.19x102 6.419 x 103 641.9x101 6419. 6419

2. Write 0.000654 in scientific notation. Examples: 2. Write 0.000654 in scientific notation. power of 10 decimal after first nonzero digit 0.000654 0.00654 x 10-1 0.0654 x 10-2 0.654 x 10-3 6.54 x 10-4

CALCULATIONS WITH SCIENTIFIC NOTATION To perform multiplication or division with scientific notation: Change numbers to exponential form. Multiply or divide coefficients. Add exponents if multiplying, or subtract exponents if dividing. If needed, reconstruct answer in standard exponential form.

Multiply coefficients Convert to exponential form Example 1: Multiply 30,000 by 600,000 Add exponents Reconstruct answer Multiply coefficients Convert to exponential form 9 (3 x 104) (6 x 105) = 18 x 10 1.8 x 1010

Convert to exponential form Example 2: Divided 30,000 by 0.006 Subtract exponents Reconstruct answer Divide coefficients Convert to exponential form 4 – (-3) (3 x 104) 7 = 0.5 x 10 (6 x 10-3) 5 x 106

Follow-up Problems: (5.5x103)(3.1x105) = 17.05x108 = 1.7x109

Follow-up Problems: (8.75x1014)(3.6x108) = 31.5x1022 = 3.2x1023

Is this measurement precise? ACCURACY & PRECISION Precision is the reproducibility of a measurement compared to other similar measurements. Precision describes how close measurements are to one another. Precision is affected by random errors. Is this measurement precise? Avg mass = 3.12± 0.01 g This measurement has high precision because the deviation of multiple trials is small.

ACCURACY & PRECISION Is this measurement accurate? Accuracy is the closeness of a measurement to an accepted value (external standard). Accuracy describes how true a measurement is. Accuracy is affected by systematic errors. Avg mass = 3.12± 0.01 g True mass = 3.03 g Accuracy cannot be determined without knowledge of the accepted value. This measurement has low accuracy because the deviation from true value is large.

ACCURACY & PRECISION Poor precision Good precision Good accuracy Poor accuracy Poor precision Good precision Poor accuracy Good accuracy

ACCURACY & PRECISION Two types of error can affect measurements: Systematic errors: those errors that are controllable, and cause measurements to be either higher or lower than the actual value. Random errors: those errors that are uncontrollable, and cause measurements to be both higher and lower than the average value.

ERROR IN MEASUREMENTS Two kinds of numbers are used in science: Counted or defined: exact numbers; have no uncertainty Measured: are subject to error; have uncertainty Every measurement has uncertainty because of instrument limitations and human error.

ERROR IN MEASUREMENTS certain uncertain 8.65 8.6 uncertain certain The last digit in any measurement is the estimated one. What is this measurement? What is this measurement?

RECORDING MEASUREMENTS TO THE PROPER NO OF DIGITS What is the correct value for each measurement? a) 28ml (1 certain, 1 uncertain) b) 28.2ml (2 certain, 1 uncertain) c) 28.31ml (3 certain, 1 uncertain)

SIGNIFICANT FIGURES RULES Significant figures rules are used to determine which digits are significant and which are not. Significant figures are the certain and uncertain digits in a measurement. All non-zero digits are significant. All sandwiched zeros are significant. Leading zeros (before or after a decimal) are NOT significant. Trailing zeros (after a decimal) are significant. 0 . 0 0 4 0 0 4 5 0 0

Examples: Determine the number of significant figures in each of the following measurements. 93.500 g 461 cm 3 sig figs 5 sig figs 0.006 m 1025 g 1 sig fig 4 sig figs 0.705 mL 5500 km 2 sig figs 3 sig figs

ROUNDING OFF NUMBERS 51.234 Round to 3 sig figs 1.875377 If rounded digit is less than 5, the digit is dropped. 51.234 Round to 3 sig figs 1.875377 Round to 4 sig figs Less than 5 Less than 5

ROUNDING OFF NUMBERS 51.369 Round to 3 sig figs 4 5.4505 1 If rounded digit is equal to or more than 5, the digit is increased by 1. 51.369 Round to 3 sig figs 4 5.4505 1 Round to 4 sig figs More than 5 Equal to 5

SIGNIFICANT FIGURES & CALCULATIONS The results of a calculation cannot be more precise than the least precise measurement. In multiplication or division, the answer must contain the same number of significant figures as in the measurement that has the least number of significant figures. For addition and subtraction, the answer must have the same number of decimal places as there are in the measurement with the fewest decimal places.

MULTIPLICATION & DIVISION Calculator answer 4 sig figs 3 sig figs (9.2)(6.80)(0.3744) = 23.4225 2 sig figs The answer should have two significant figures because 9.2 is the number with the fewest significant figures. The correct answer is 23

ADDITION & SUBTRACTION Add 83.5 and 23.28 83.5 23.28 Least precise number Calculator answer 106.78 106.8 Correct answer

Example 1: 5.008 + 16.2 + 13.48 = 34.688 34.7 Least precise number Round to

Example 2: 3 sig figs 6.1788 6.2 2 sig figs Round to

SI UNITS Measurements are made by scientists to determine size, length and other properties of matter. For measurements to be useful, a measurement standard must be used. A standard is an exact quantity that people agree to use for comparison. SI is the standard system of measurement used worldwide by scientists.

SI (METRIC) BASE UNITS Quantity Measured Metric Units Symbol English Length Meter m yd Mass Kilogram kg lb Time Seconds s Temperature Kelvin K F Amount of substance Mole mol

Basic Units of Measurement The kilogram is a measure of mass, which is different from weight. The mass of an object is a measure of the quantity of matter within it. The weight of an object is a measure of the gravitational pull on that matter. Consequently, weight depends on gravity while mass does not.

Derived Units A derived unit is formed from other units. Many units of volume, a measure of space, are derived units. Any unit of length, when cubed (raised to the third power), becomes a unit of volume. Cubic meters (m3), cubic centimeters (cm3), and cubic millimeters (mm3) are all units of volume. Change to equation objects

DERIVED UNITS In addition to the base units, several derived units are commonly used in SI system. Quantity Measured Units Symbol Volume Liter L Density grams/cc g/cm3

SI PREFIXES Common prefixes are used with the base units to indicate the multiple of ten that the unit represents. The SI system of units is easy to use because it is based on multiples of ten. SI Prefixes Prefixes Symbol Multiplying factor mega- M 1,000,000 kilo- k 1000 centi- c 0.01 milli- m 0.001 micro-  0.000,001 106 103 10-2 10-3 10-6

SI UNITS & PREFIXES SI system used a common set of prefixes for use with the base units. Base Unit micro 106 milli 103 kilo 103 mega 106 10 10 10 10 10 10 deci centi Smaller units Larger units

SI CONVERSION FACTORS Base Unit micro 106 kilo 103 mega 106 milli 103 10 10 10 10 10 10 deci centi 1 m = 103 mm or 1 mm = 103 m 1 mm = 103 m or 1 m = 103 mm

SI PREFIXES 100000 or 105 How many mm are in a cm? How many cm are in a km? 10x10x10x10x10 10

Prefix Multipliers Choose the prefix multiplier that is most convenient for a particular measurement. Pick a unit similar in size to (or smaller than) the quantity you are measuring. A short chemical bond is about 1.2 × 10–10 m. Which prefix multiplier should you use? The most convenient one is probably the picometer. Chemical bonds measure about 120 pm. Change to equation object

CONVERSION FACTORS Many problems in chemistry and related fields require a change of units. Any unit can be converted into another by use of the appropriate conversion factor. Any equality in units can be written in the form of a fraction called a conversion factor. For example: Metric-Metric Factor Equality 1 m = 100 cm 1 m 100 cm 100 cm 1 m Conversion Factors or

Metric-English Factor CONVERSION FACTORS Metric-English Factor Equality 1 kg = 2.20 lb 1 kg 2.20 lb 2.20 lb 1 kg Conversion Factors or Percentage Factor Sometimes a conversion factor is given as a percentage. For example: Percent quantity: 18% body fat by mass Conversion Factors 18 kg body fat 100 kg body mass 100 kg body mass 18 kg body fat or

CONVERSION OF UNITS final unit beginning unit Conversion factor Problems involving conversion of units and other chemistry problems can be solved using the following step-wise method: 3. Write the conversion factor for each units change in your plan. 4. Set up the problem by arranging cancelling units in the numerator and denominator of the steps involved. Plan a sequence of steps to convert the initial unit to the final unit. 1. Determine the intial unit given and the final unit needed. final unit beginning unit Conversion factor

Metric-English factor Example 1: Convert 164 lb to kg (1 kg = 2.20 lb) Step 1: Given: 164 lb Need: kg Metric-English factor lb kg Step 2: 1 kg 2.20 lb 2.20 lb 1 kg Step 3: or Step 4:

Example 2: The thickness of a book is 2.5 cm. What is this measurement in mm? Step 1: Given: 2.5 cm Need: mm Metric-Metric factor cm mm Step 2: 1 cm 10 mm 10 mm 1 cm Step 3: or Step 4:

English-English factor Metric-English factor Example 3: How many centimeters are in 2.0 ft? (1 in=2.54 cm) Step 1: Given: 2.0 ft Need: cm English-English factor Metric-English factor ft in cm Step 2: 1 ft 12 in 1 in 2.54 cm Step 3: and 61 cm 60.96 cm Step 4:

Given: 1.75 lb bronze Need: g of copper English-Metric factor Example 4: Bronze is 80.0% by mass copper and 20.0% by mass tin. A sculptor is preparing to case a figure that requires 1.75 lb of bronze. How many grams of copper are needed for the brass figure (1lb = 454g)? Step 1: Given: 1.75 lb bronze Need: g of copper lb brz English-Metric factor g brz Percentage factor g Cu Step 2:

Example 4: 1 lb 454 g 80.0 g Cu 100 g brz 454 g 1 lb 80.0 g Cu Step 3: and 1.75 lb brz x 454 g 1 lb 80.0 g Cu 100 g brz Step 4: x = 635.6 g = 636 g

VOLUME Volume is the amount of space an object occupies. Common units are cm3 or liter (L) and milliliter (mL). 1 L = 1000 mL 1 mL = 1 cm3

VOLUME Volume of various regular shapes can be calculated as follows: Cube V = s x s x s Rect. V = l x w x h Cylinder V = π x r2 x h Sphere V =4/3 πr3

Which has greatest density? Density is mass per unit volume of a material. Common units are g/cm3 (solids) or g/mL (liquids). Density is directly related to the mass of an object. Density is indirectly related to the volume of an object. Which has greatest density?

Example 1: A copper sample has a mass of 44.65 g and a volume of 5.0 mL. What is the density of copper? m = 44.65 g v = 5.0 mL d = ??? = 8.93 g/mL = 8.9 g/mL Round to 2 sig figs

Example 2: A silver bar with a volume of 28.0 cm3 has a mass of 294 g. What is the density of this bar? m = 294 g v = 28.0 mL d = ??? = 10.5 g/mL 3 sig figs

Example 3: If the density of gold is 19.3 g/cm3, how many grams does a 5.00 cm3 nugget weigh? Step 1: Given: 5.00 cm3 Need: g density Step 2: cm3 g Step 3: 5.00 cm3 = 96.5 g

English-Metric factor Example 4: If the density of milk is 1.04 g/mL, what is the mass of 0.50 qt of milk? (1L = 1.06 qt) Step 1: Given: 0.5 qt Need: g English-Metric factor Step 2: qt Metric-Metric factor density L mL g 1 L 1.06 qt 103 mL 1 L 1.04 g 1 mL Step 3: and and Step 4: = 490.57 g = 490 g

Example 5: What volume of mercury has a mass of 60.0 g if its density is 13.6 g/mL? 1 13.6 mL 4.41 60.0 g g inverse of density

IS UNIT CONVERSION IMPORTANT? In 1999 Mars Climate orbiter was lost in space because engineers failed to make a simple conversion from English units to metric units, an embarrassing lapse that sent the $125 million craft fatally close to the Martian surface. Further investigation showed that engineers at Lockheed Martin, which built the aircraft, calculated navigational measurements in English units. When NASA’s JPL engineers received the data, they assumed the information was in metric units, causing the confusion.

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